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Theorem quad2 23367
Description: The quadratic equation, without specifying the particular branch  D to the square root. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
quad.a  |-  ( ph  ->  A  e.  CC )
quad.z  |-  ( ph  ->  A  =/=  0 )
quad.b  |-  ( ph  ->  B  e.  CC )
quad.c  |-  ( ph  ->  C  e.  CC )
quad.x  |-  ( ph  ->  X  e.  CC )
quad2.d  |-  ( ph  ->  D  e.  CC )
quad2.2  |-  ( ph  ->  ( D ^ 2 )  =  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )
Assertion
Ref Expression
quad2  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( X  =  ( ( -u B  +  D )  /  (
2  x.  A ) )  \/  X  =  ( ( -u B  -  D )  /  (
2  x.  A ) ) ) ) )

Proof of Theorem quad2
StepHypRef Expression
1 2cn 10602 . . . . . . . 8  |-  2  e.  CC
2 quad.a . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
3 mulcl 9565 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
41, 2, 3sylancr 661 . . . . . . 7  |-  ( ph  ->  ( 2  x.  A
)  e.  CC )
5 quad.x . . . . . . 7  |-  ( ph  ->  X  e.  CC )
64, 5mulcld 9605 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  A )  x.  X
)  e.  CC )
7 quad.b . . . . . 6  |-  ( ph  ->  B  e.  CC )
86, 7addcld 9604 . . . . 5  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  +  B
)  e.  CC )
98sqcld 12290 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  e.  CC )
10 quad2.d . . . . 5  |-  ( ph  ->  D  e.  CC )
1110sqcld 12290 . . . 4  |-  ( ph  ->  ( D ^ 2 )  e.  CC )
129, 11subeq0ad 9932 . . 3  |-  ( ph  ->  ( ( ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 )  -  ( D ^ 2 ) )  =  0  <->  (
( ( ( 2  x.  A )  x.  X )  +  B
) ^ 2 )  =  ( D ^
2 ) ) )
135sqcld 12290 . . . . . . 7  |-  ( ph  ->  ( X ^ 2 )  e.  CC )
142, 13mulcld 9605 . . . . . 6  |-  ( ph  ->  ( A  x.  ( X ^ 2 ) )  e.  CC )
157, 5mulcld 9605 . . . . . . 7  |-  ( ph  ->  ( B  x.  X
)  e.  CC )
16 quad.c . . . . . . 7  |-  ( ph  ->  C  e.  CC )
1715, 16addcld 9604 . . . . . 6  |-  ( ph  ->  ( ( B  x.  X )  +  C
)  e.  CC )
1814, 17addcld 9604 . . . . 5  |-  ( ph  ->  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  e.  CC )
19 0cnd 9578 . . . . 5  |-  ( ph  ->  0  e.  CC )
20 4cn 10609 . . . . . 6  |-  4  e.  CC
21 mulcl 9565 . . . . . 6  |-  ( ( 4  e.  CC  /\  A  e.  CC )  ->  ( 4  x.  A
)  e.  CC )
2220, 2, 21sylancr 661 . . . . 5  |-  ( ph  ->  ( 4  x.  A
)  e.  CC )
2320a1i 11 . . . . . 6  |-  ( ph  ->  4  e.  CC )
24 4ne0 10628 . . . . . . 7  |-  4  =/=  0
2524a1i 11 . . . . . 6  |-  ( ph  ->  4  =/=  0 )
26 quad.z . . . . . 6  |-  ( ph  ->  A  =/=  0 )
2723, 2, 25, 26mulne0d 10197 . . . . 5  |-  ( ph  ->  ( 4  x.  A
)  =/=  0 )
2818, 19, 22, 27mulcand 10178 . . . 4  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( 4  x.  A )  x.  0 )  <->  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  =  0 ) )
296sqcld 12290 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X ) ^ 2 )  e.  CC )
306, 7mulcld 9605 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  x.  B
)  e.  CC )
31 mulcl 9565 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( ( ( 2  x.  A )  x.  X )  x.  B
)  e.  CC )  ->  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B
) )  e.  CC )
321, 30, 31sylancr 661 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) )  e.  CC )
332, 16mulcld 9605 . . . . . . . . 9  |-  ( ph  ->  ( A  x.  C
)  e.  CC )
34 mulcl 9565 . . . . . . . . 9  |-  ( ( 4  e.  CC  /\  ( A  x.  C
)  e.  CC )  ->  ( 4  x.  ( A  x.  C
) )  e.  CC )
3520, 33, 34sylancr 661 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( A  x.  C )
)  e.  CC )
3629, 32, 35addassd 9607 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) )  +  ( 4  x.  ( A  x.  C ) ) )  =  ( ( ( ( 2  x.  A )  x.  X
) ^ 2 )  +  ( ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) )  +  ( 4  x.  ( A  x.  C )
) ) ) )
377sqcld 12290 . . . . . . . 8  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
3829, 32addcld 9604 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X ) ^
2 )  +  ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) ) )  e.  CC )
3937, 38, 35pnncand 9961 . . . . . . 7  |-  ( ph  ->  ( ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A )  x.  X ) ^
2 )  +  ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) ) ) )  -  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  =  ( ( ( ( ( 2  x.  A )  x.  X ) ^ 2 )  +  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )  +  ( 4  x.  ( A  x.  C
) ) ) )
404, 5sqmuld 12304 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X ) ^ 2 )  =  ( ( ( 2  x.  A
) ^ 2 )  x.  ( X ^
2 ) ) )
41 sq2 12246 . . . . . . . . . . . . 13  |-  ( 2 ^ 2 )  =  4
4241a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2 ^ 2 )  =  4 )
432sqvald 12289 . . . . . . . . . . . 12  |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
4442, 43oveq12d 6288 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( A ^ 2 ) )  =  ( 4  x.  ( A  x.  A
) ) )
45 sqmul 12213 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( A ^
2 ) ) )
461, 2, 45sylancr 661 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( A ^
2 ) ) )
4723, 2, 2mulassd 9608 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 4  x.  A )  x.  A
)  =  ( 4  x.  ( A  x.  A ) ) )
4844, 46, 473eqtr4d 2505 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 4  x.  A )  x.  A ) )
4948oveq1d 6285 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A ) ^
2 )  x.  ( X ^ 2 ) )  =  ( ( ( 4  x.  A )  x.  A )  x.  ( X ^ 2 ) ) )
5022, 2, 13mulassd 9608 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  A )  x.  ( X ^ 2 ) )  =  ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) ) )
5140, 49, 503eqtrrd 2500 . . . . . . . 8  |-  ( ph  ->  ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  =  ( ( ( 2  x.  A
)  x.  X ) ^ 2 ) )
5222, 15, 16adddid 9609 . . . . . . . . 9  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( B  x.  X
)  +  C ) )  =  ( ( ( 4  x.  A
)  x.  ( B  x.  X ) )  +  ( ( 4  x.  A )  x.  C ) ) )
53 2t2e4 10681 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  2 )  =  4
5453oveq1i 6280 . . . . . . . . . . . . . . . 16  |-  ( ( 2  x.  2 )  x.  A )  =  ( 4  x.  A
)
551a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  2  e.  CC )
5655, 55, 2mulassd 9608 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 2  x.  2 )  x.  A
)  =  ( 2  x.  ( 2  x.  A ) ) )
5754, 56syl5eqr 2509 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 4  x.  A
)  =  ( 2  x.  ( 2  x.  A ) ) )
5857oveq1d 6285 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 4  x.  A )  x.  B
)  =  ( ( 2  x.  ( 2  x.  A ) )  x.  B ) )
5955, 4, 7mulassd 9608 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 2  x.  ( 2  x.  A
) )  x.  B
)  =  ( 2  x.  ( ( 2  x.  A )  x.  B ) ) )
6058, 59eqtrd 2495 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 4  x.  A )  x.  B
)  =  ( 2  x.  ( ( 2  x.  A )  x.  B ) ) )
6160oveq1d 6285 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( ( 2  x.  ( ( 2  x.  A )  x.  B ) )  x.  X ) )
624, 7mulcld 9605 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 2  x.  A )  x.  B
)  e.  CC )
6355, 62, 5mulassd 9608 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 2  x.  ( ( 2  x.  A )  x.  B
) )  x.  X
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  B )  x.  X ) ) )
6461, 63eqtrd 2495 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  B )  x.  X ) ) )
6522, 7, 5mulassd 9608 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( ( 4  x.  A )  x.  ( B  x.  X ) ) )
664, 7, 5mul32d 9779 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  B )  x.  X
)  =  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )
6766oveq2d 6286 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( 2  x.  A )  x.  B
)  x.  X ) )  =  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )
6864, 65, 673eqtr3d 2503 . . . . . . . . . 10  |-  ( ph  ->  ( ( 4  x.  A )  x.  ( B  x.  X )
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )
6923, 2, 16mulassd 9608 . . . . . . . . . 10  |-  ( ph  ->  ( ( 4  x.  A )  x.  C
)  =  ( 4  x.  ( A  x.  C ) ) )
7068, 69oveq12d 6288 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( B  x.  X
) )  +  ( ( 4  x.  A
)  x.  C ) )  =  ( ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )  +  ( 4  x.  ( A  x.  C
) ) ) )
7152, 70eqtrd 2495 . . . . . . . 8  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( B  x.  X
)  +  C ) )  =  ( ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )  +  ( 4  x.  ( A  x.  C
) ) ) )
7251, 71oveq12d 6288 . . . . . . 7  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  +  ( ( 4  x.  A )  x.  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B
) )  +  ( 4  x.  ( A  x.  C ) ) ) ) )
7336, 39, 723eqtr4rd 2506 . . . . . 6  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  +  ( ( 4  x.  A )  x.  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) )  -  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )
7422, 14, 17adddid 9609 . . . . . 6  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  +  ( ( 4  x.  A
)  x.  ( ( B  x.  X )  +  C ) ) ) )
75 binom2 12265 . . . . . . . . 9  |-  ( ( ( ( 2  x.  A )  x.  X
)  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( ( ( ( ( 2  x.  A )  x.  X ) ^ 2 )  +  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )  +  ( B ^
2 ) ) )
766, 7, 75syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( ( ( ( ( 2  x.  A )  x.  X ) ^ 2 )  +  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )  +  ( B ^
2 ) ) )
7738, 37addcomd 9771 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) )  +  ( B ^ 2 ) )  =  ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) ) )
7876, 77eqtrd 2495 . . . . . . 7  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) ) )
79 quad2.2 . . . . . . 7  |-  ( ph  ->  ( D ^ 2 )  =  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )
8078, 79oveq12d 6288 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  +  B ) ^
2 )  -  ( D ^ 2 ) )  =  ( ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) )  -  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )
8173, 74, 803eqtr4d 2505 . . . . 5  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 )  -  ( D ^ 2 ) ) )
8222mul01d 9768 . . . . 5  |-  ( ph  ->  ( ( 4  x.  A )  x.  0 )  =  0 )
8381, 82eqeq12d 2476 . . . 4  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( 4  x.  A )  x.  0 )  <->  ( (
( ( ( 2  x.  A )  x.  X )  +  B
) ^ 2 )  -  ( D ^
2 ) )  =  0 ) )
8428, 83bitr3d 255 . . 3  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( ( ( ( ( 2  x.  A
)  x.  X )  +  B ) ^
2 )  -  ( D ^ 2 ) )  =  0 ) )
856, 7subnegd 9929 . . . . 5  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  -  -u B
)  =  ( ( ( 2  x.  A
)  x.  X )  +  B ) )
8685oveq1d 6285 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B ) ^ 2 )  =  ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 ) )
8786eqeq1d 2456 . . 3  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B ) ^
2 )  =  ( D ^ 2 )  <-> 
( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( D ^ 2 ) ) )
8812, 84, 873bitr4d 285 . 2  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( ( ( ( 2  x.  A )  x.  X )  -  -u B ) ^ 2 )  =  ( D ^ 2 ) ) )
897negcld 9909 . . . 4  |-  ( ph  -> 
-u B  e.  CC )
906, 89subcld 9922 . . 3  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  -  -u B
)  e.  CC )
91 sqeqor 12264 . . 3  |-  ( ( ( ( ( 2  x.  A )  x.  X )  -  -u B
)  e.  CC  /\  D  e.  CC )  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B ) ^
2 )  =  ( D ^ 2 )  <-> 
( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  \/  ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D ) ) )
9290, 10, 91syl2anc 659 . 2  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B ) ^
2 )  =  ( D ^ 2 )  <-> 
( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  \/  ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D ) ) )
936, 89, 10subaddd 9940 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  <-> 
( -u B  +  D
)  =  ( ( 2  x.  A )  x.  X ) ) )
9489, 10addcld 9604 . . . . . 6  |-  ( ph  ->  ( -u B  +  D )  e.  CC )
95 2ne0 10624 . . . . . . . 8  |-  2  =/=  0
9695a1i 11 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
9755, 2, 96, 26mulne0d 10197 . . . . . 6  |-  ( ph  ->  ( 2  x.  A
)  =/=  0 )
9894, 4, 5, 97divmuld 10338 . . . . 5  |-  ( ph  ->  ( ( ( -u B  +  D )  /  ( 2  x.  A ) )  =  X  <->  ( ( 2  x.  A )  x.  X )  =  (
-u B  +  D
) ) )
99 eqcom 2463 . . . . 5  |-  ( X  =  ( ( -u B  +  D )  /  ( 2  x.  A ) )  <->  ( ( -u B  +  D )  /  ( 2  x.  A ) )  =  X )
100 eqcom 2463 . . . . 5  |-  ( (
-u B  +  D
)  =  ( ( 2  x.  A )  x.  X )  <->  ( (
2  x.  A )  x.  X )  =  ( -u B  +  D ) )
10198, 99, 1003bitr4g 288 . . . 4  |-  ( ph  ->  ( X  =  ( ( -u B  +  D )  /  (
2  x.  A ) )  <->  ( -u B  +  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10293, 101bitr4d 256 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  <-> 
X  =  ( (
-u B  +  D
)  /  ( 2  x.  A ) ) ) )
10389, 10negsubd 9928 . . . . 5  |-  ( ph  ->  ( -u B  +  -u D )  =  (
-u B  -  D
) )
104103eqeq1d 2456 . . . 4  |-  ( ph  ->  ( ( -u B  +  -u D )  =  ( ( 2  x.  A )  x.  X
)  <->  ( -u B  -  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10510negcld 9909 . . . . 5  |-  ( ph  -> 
-u D  e.  CC )
1066, 89, 105subaddd 9940 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D 
<->  ( -u B  +  -u D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10789, 10subcld 9922 . . . . . 6  |-  ( ph  ->  ( -u B  -  D )  e.  CC )
108107, 4, 5, 97divmuld 10338 . . . . 5  |-  ( ph  ->  ( ( ( -u B  -  D )  /  ( 2  x.  A ) )  =  X  <->  ( ( 2  x.  A )  x.  X )  =  (
-u B  -  D
) ) )
109 eqcom 2463 . . . . 5  |-  ( X  =  ( ( -u B  -  D )  /  ( 2  x.  A ) )  <->  ( ( -u B  -  D )  /  ( 2  x.  A ) )  =  X )
110 eqcom 2463 . . . . 5  |-  ( (
-u B  -  D
)  =  ( ( 2  x.  A )  x.  X )  <->  ( (
2  x.  A )  x.  X )  =  ( -u B  -  D ) )
111108, 109, 1103bitr4g 288 . . . 4  |-  ( ph  ->  ( X  =  ( ( -u B  -  D )  /  (
2  x.  A ) )  <->  ( -u B  -  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
112104, 106, 1113bitr4d 285 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D 
<->  X  =  ( (
-u B  -  D
)  /  ( 2  x.  A ) ) ) )
113102, 112orbi12d 707 . 2  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B )  =  D  \/  ( ( ( 2  x.  A
)  x.  X )  -  -u B )  = 
-u D )  <->  ( X  =  ( ( -u B  +  D )  /  ( 2  x.  A ) )  \/  X  =  ( (
-u B  -  D
)  /  ( 2  x.  A ) ) ) ) )
11488, 92, 1133bitrd 279 1  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( X  =  ( ( -u B  +  D )  /  (
2  x.  A ) )  \/  X  =  ( ( -u B  -  D )  /  (
2  x.  A ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    = wceq 1398    e. wcel 1823    =/= wne 2649  (class class class)co 6270   CCcc 9479   0cc0 9481    + caddc 9484    x. cmul 9486    - cmin 9796   -ucneg 9797    / cdiv 10202   2c2 10581   4c4 10583   ^cexp 12148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-n0 10792  df-z 10861  df-uz 11083  df-seq 12090  df-exp 12149
This theorem is referenced by:  quad  23368  dcubic2  23372  dquartlem1  23379
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